I have been given the equation $v^{\prime\prime}(t) + t\sin(v(t)) = 0$. The problem is to show that the solution $v(t) \equiv \pi$ is unstable and that the solution $v(t) \equiv 0$ is asymptotically stable. When asked for a hint, the professor made special note of the fact that \begin{equation} v^{\prime\prime}(t) + t\sin(v(t)) = 0 \implies 0 \leq |v^\prime(t) - v^\prime(0)| \leq \Big|-\int_0^t s\sin(v(s))ds\Big| \leq \frac{1}{2}t^2.\end{equation} However, this doesn't actually give a lower bound on $v^\prime(t)$, because there is nothing forcing $v^\prime(0) > 0$. How else could we go about this? Or is there something I'm missing?
More generally, is there a general technique (I know such a theorem doesn't exist) that tends to allow one to determine if a particular solution of a system is unstable, stable, or asymptotically stable? If so, what is it?